Fractal geometry has been used over the past 15 years as an analytical tool to mathematically describe curves, surfaces or volumes of a certain "roughness" when they appear equally rough or pleated under a wide range of scales. These scale invariant structures are described in terms of their fractal dimensions (called the Hausdorff dimension "d") by the Mandelbrot equation:
______________________________________ N.sub.n.sup.d = constant (l.sub.o.sup.d) (Ref Mandelbrot, B., "Les Objects Fractals" - Flammarion Pans 1995). ______________________________________ Where .sub.n is a gauge of measure N is the count obtained by measuring the medium with the gauge n l.sub.o is the characteristic length .sup.d (integer or noninteger) is the Hausdorff (fractal) dimension (dCR)
In the case of Euclidean geometry, if a segment is gauged by a half scale of its length (n=1/2) its N count is given by 2.sup.1, but for the equivalent square gauged in the same way, d=2 and its N count (or measure) is given by 2.sup.2, and for a cube d=3 and N=2.sup.3 =8, the Mandelbrot equation gives EQU 8(1/2).sup.3 =1.sup.3
In the case of fractal geometry, take the example of the Von Koch curve (FIG. 1) to generalise the application of the Mandelbrot equation. To form the Von Koch curve, the generator of the curve is taken from the segment lo. The segment is cut into three parts, the centre part is removed and two segments of equal length are added to form l.sub.1, which is called the generator. In this case the length l.sub.n of the curve of order n is l.sub.(n) -l.sub.o .eta..sup.n-1 and the pattern tends to a limit (the d fractal) where d=log.sub.3 4=1.26186 ##STR1## Starting with other generators, and using the same principle there is provided for example ##STR2## The construction process can be further generalised using for example the following generators in a two dimensional plane ##STR3##
This concept is particularly useful in organic chemistry and has been used to model the polymerisation of various monomeric units when stereospecific polymerisation is involved.
Advanced fractal geometry concepts like multifractals applied to random walk are also routinely used to describe simple step by step polymerisation reactions when lateral reticulation occurs.
It is known that the solubility of smaller molecules is totally different to the solubility or larger molecules. The solubility of polymers varies as a function of the length of the polymer, which means that it is possible to precipitate the larger molecules before the smaller molecules are precipitated. This means that the compound structure will be determined by means of solubility step reactions.
It is an object of this invention to provide polymers having a fractal configuration, and to provide methods of preparation of fractal polymers of desired fractal dimensions.
It is another object of this invention to provide fractal polymers which, as a consequence of their fractal configuration, have superior properties for the adsorption of materials.
It is a further object of this invention to provide fractal polymers which are useful as an adjunct to the filtration or separation of compounds by conventional filters or membranes.
These are other objects of the invention will become more apparent from the following description and illustrations.